#NaturalLogarithm #Science #Engineering #LogarithmBase10 #Mathematics
When it comes to mathematics, especially in the fields of science and engineering, the natural logarithm (log base e) is more commonly used than the logarithm base 10. But why is that the case? Let’s delve into the reasons behind this phenomenon.
##Understanding Logarithms
Before we dive into why the natural logarithm is preferred over the logarithm base 10, let’s first understand what logarithms are in general. Logarithms are essentially the inverse operation of exponentiation. In simpler terms, they help us answer questions like “What power must we raise a certain number to get another number?”
##Why is the Natural Logarithm (log base e) Preferred?
There are several key reasons why the natural logarithm is favored over the logarithm base 10 in many areas of science and engineering:
1. **Euler’s Number (e)**: The natural logarithm is based on Euler’s Number (e), which is an irrational number approximately equal to 2.71828. This number has unique properties that make it particularly useful in mathematical calculations.
2. **Simplicity in Calculations**: When dealing with mathematical equations and calculations in science and engineering, the natural logarithm often leads to simpler and more elegant solutions compared to the logarithm base 10.
3. **Derivatives and Integrals**: The natural logarithm has special properties when it comes to calculus, making it easier to differentiate and integrate functions involving e compared to log base 10.
4. **Growth and Decay Processes**: In many real-world applications, such as population growth, radioactive decay, and exponential growth, the natural logarithm naturally emerges as the more appropriate choice due to its unique properties.
5. **Universal Applicability**: The natural logarithm is not limited to base 10 but can be applied universally to a wide range of mathematical problems. This versatility makes it a go-to option in various scientific disciplines.
##Examples of Natural Logarithm Usage
To further illustrate the importance of the natural logarithm in science and engineering, let’s take a look at some specific examples where it is commonly used:
– **Electrical Engineering**: In electrical circuits, the natural logarithm is used to model transient responses and analyze the behavior of systems over time.
– **Physics**: In physics, the natural logarithm often appears in equations related to exponential decay, radioactive half-life, and oscillatory motion.
– **Chemistry**: In chemical kinetics, the natural logarithm is employed to determine reaction rates and rate constants.
– **Biology**: Biologists utilize the natural logarithm to study population growth, enzyme kinetics, and other biological processes that exhibit exponential behavior.
##Conclusion
In conclusion, the natural logarithm (log base e) is more commonly used than the logarithm base 10 in many areas of science and engineering due to its unique properties, simplicity in calculations, and universal applicability. By understanding the advantages of the natural logarithm and its relevance in various scientific disciplines, we can appreciate its significance in shaping the foundation of modern mathematics and technology. So next time you come across the natural logarithm in your scientific endeavors, remember the reasons behind its widespread usage and embrace its power in solving complex problems with elegance and precision. 🌟
So, the next time you encounter the natural logarithm in your scientific or engineering studies, appreciate its significance and versatility in simplifying complex calculations and analyses. Happy calculating! 🚀
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Keyword: natural logarithm, science and engineering, Euler’s number, applications of logarithms
10 is an artifical number, the only reason we really use it is we have 10 fingers.
e is the “natural” choice because e^x is its own derivative. The only other natural choice is 2 for computing and information theory.
There’s a reason it’s called the “natural” logarithm, and that’s because it’s base e and logarithms are related to exponents (the inverse). e is important and “natural” in math because e^x is it’s own derivative, and that’s an EXTREMELY useful fact in much of math. And because logarithms are the inverse to exponents, and e^x is such an important exponent, ln is such an important logarithm.Â
Note, there’s nothing magical about it, some number HAD to be, it is guaranteed, and e is that number. Â
Also note that log 10 is still widely used outside of mathematical equations. If you ever see a log-scaled graph and it has units, that’s going to be a log 10, not ln. As important as e is to math, it is extremely inconvenient to present your numerical results in base e rather than base 10, lol.
Its because of the exponential function. We like exp because:
1) its defining property is that exp(a+b)=exp(a)exp(b) usually very neat
2) its derivitive is itself, its a derivitive fix point.
As a consequence of that when we have differential equations that are something on the line of: the derivitive of the function is proportional to itself we get exp as a solution. And we often deal with diff equations like that.
What if we used 10 instead of e? Lets look at the derivitive of 10^(x). We can rewrite a power like a^(b) = e^b(lna) = e^(lna)^(b) = a^(b). So we can take the derivitive of 10^(x) = e^(xln(10)) = exp(x × ln(10)) and its derivitive is exp(ln10 × x) × ln(10) = 10^x × ln 10. (And its not like ln is avoidable here, the whole point is that you can redefine powers with a workable function.)
As you can see using anything other than e^(x) for an exponential is an unnecessary complication. e^(x) is just more convenient function to work with and so its inverse also shows up more frequently than others base logarithms. Thats why ln is called the natural base log or natural log, thats what the n stands for. Its in a sense more natural to use as anything other compared to it is just a handicap.
Of course in computer science base 2 logs can show up quite frequently if you insist on thinking in terms of bits.
Something also not mentioned here is that logarithm base choice hardly ever matters. We have the change of base formula to move between them if it’s ever necessary to use one in particular.
A really common phenomenon in many branches of science is that a quantity of something may change over time depending on how much of that thing there currently is. For example, populations of bacteria, the spread of disease, and returns on investment portfolios can all follow this rule. (The more bacteria present in a sample, the more they multiply, so the more bacteria are present, and so on.)
In these instances, you can predict future values using the exponential growth formula: y = C * e ^ (R * T). In this formula, C and R are known constants for your problem, T is the amount of time elapsed, and ‘e’ is Euler’s number, approximately equal to 2.718.
If you want to isolate the exponent to do something with it, you use a logarithm of the same base reverse the exponentiation. For example, the equation y = e ^ T could also be written as ln(y) = T, where ln is the logarithm base ‘e’. Because ‘e’ is used in the exponential growth formula, the log base e appears frequently in these contexts.
But why the number ‘e’ specifically. What is so special about 2.718? It has to do with something called compounding.
Imagine you have found a bank that is willing to give you 100% interest on your account every year. You put in $100.00 on January 1st and then on December 31st the bank gives you your 100% interest, giving you a total of $200.00.
Now imagine that instead of paying 100% interest once at the end of the year, the bank decides to give you 50% interest twice a year instead. On May 30th, the bank gives you your 50% interest on your $100.00 giving you $150.00 in your account. Then on December 31st the bank gives you 50% interest on your $150.00 giving you $225.00 in your account.
This is called compounding. The bank is still paying the same 100% interest rate, but the money the bank paid you in May is now also generating interest for you in December, giving you more at the end of the year. But we don’t have to stop there, if the bank compounds more frequently, you will make more money, up to a point:
1 payment of 100%: $200.00
2 payments of 50%: $225.00
4 payments of 25%: $244.14
8 payments of 12.5%: $256.57
16 payments of 6.25%: $263.79
32 payments of 3.125%: $267.69
…
1,000,000 payments of 0.0001%: $271.82
It turns out, that if the bank compounds your money as frequently as possible, at the end of the year, you will wind up with $271.82 in your account, or ‘e’ times the amount that you put in.
In the natural world, this is exactly what happens. Bacteria don’t wait a day and then double all at once, they are compounding continuously. Hence, ‘e’ becomes the natural choice for the exponential growth formula.
why base 10? there is no reason to ever have base 10 in anything actually. nothing special about 10
Something super important that hasn’t yet been said is that an association of two natural log transformed variables represents a proportional change relation to them (e.g., for every one % increase in the predictor X, the outcome increases or decreases by Y%).
https://stats.stackexchange.com/questions/244199/why-is-it-that-natural-log-changes-are-percentage-changes-what-is-about-logs-th
This isn’t the case for log10, which makes sciences like the natural log. It keeps coefficients easily interpretable.
one of the reasons is that the exp() function shows up a lot in solutions of differential equations, because it is it’s own derivative (d/dx e^x = e^x ).
To illustrate this, take the basic equation y’ = 2y. The solution of this equation is y=e^2x, because y’=2 * e^2x = 2y.
So e^x and variants of it show up as solutions of a lot of differential equations that relate higher-order derivatives to lower-order ones, since the e^x sticks around and can cancel itself out, and therefore so does ln.